Methods and systems for providing optimum decision feedback equalization of high-speed serial data links

ABSTRACT

Computationally efficient methods and related systems, for use in a test and measurement instrument, such as an oscilloscope, optimize the performance of DFEs used in a high-speed serial data link by identifying optimal DFE tap values for peak-to-peak based criteria. The optimized DFEs comply with the behavior of a model DFE set forth in the PCIE 3.0 specification.

RELATED APPLICATIONS

This application is related to, and claims the benefit of priority of U.S. Provisional Application No. 62/026,408, filed Jul. 18, 2014, and co-pending, commonly assigned, U.S. Non-Provisional patent application Ser. No. 13/326,753, filed Dec. 15, 2011, which claims the benefit of priority of Provisional Application No. 61/450,542 filed Mar. 8, 2011 and incorporates by reference herein, as if set forth in full herein, each of the above referenced applications. This application is also related to U.S. Pat. No. 8,374,231 filed Apr. 14, 2009, assigned to the same assignee as the present application, and incorporates by reference herein, as if set forth in full herein, the contents of U.S. Pat. No. 8,374,231. To the extent that any of the disclosure in U.S. Pat. No. 8,374,231 conflicts or appears to conflict with the disclosure of the present specification, the disclosure of the present specification shall take precedence and govern the resolution of any such conflict.

BACKGROUND

Generally speaking, a communication system is composed of a transmission means (e.g., transmitter) that transmits a signal, a “medium” through which the signal is transmitted and a reception means (e.g., receiver) for receiving the transmitted signal. As is known in the art, the medium may take any one of a number of physical forms, such as a copper wire, coaxial cable or in the case of wireless transmission, the air. A word used synonymous with “medium” is “transmission channel” or “channel” for short. FIG. 1 depicts a generalized communication system 1. As shown, a transmission means (abbreviated “Tx”) 2 transmits an analog waveform signal containing information through a wired channel 3 to a reception means 4 (abbreviated “Rx”). As is also known in the art, the analog waveform signal may be transmitted through the channel 3 at varying speeds ranging from a few kilobits per second to many gigabits per second (abbreviated “Gb/s”).

So-called high speed, serial data link systems involve the transmission of signals at speeds of 6 Gb/s and beyond. At such speeds, “equalization” techniques are needed, typically at the reception means or receiving end of the channel, to correct for the degradation of the signal as it passes through the channel 3 due to physical phenomena commonly referred to as channel loss, reflection, cross talk, and noise to name a few. In general, equalization involves the removal or “filtering” out of undesirable components of a signal that lead to the signal's degradation, components that were added by the phenomena discussed above as well as others.

In the high-speed, serial data link industry, standards have been promulgated that govern the type of equalization measurements to be applied. One such standard is that promulgated by the SAS-2, 6G standards committee. The standard involves an equalization method using a so-called “training sequence”. A training sequence is used to, generally speaking, adjust an equalization filter so that it can correctly remove undesirable signal components from digitized samples of a received analog waveform signal. Typically, a training sequence is derived from digitized samples (e.g., bits) of a received signal.

Determining the training sequence associated with digitized samples of a given analog waveform signal is not a trivial matter, however. U.S. Pat. No. 8,374,231 mentioned above sets forth various methods and systems (referred to as “equalization simulators”) for determining the training sequence of a signal in a high-speed, serial data link system using a real-time or sampling oscilloscope. Before a so-determined training sequence can be used to remove degradations from the signal, however, it must be aligned with re-sampled, digitized samples of the received analog waveform signal. U.S. Pat. No. 8,374,231 discloses techniques for doing so.

In effect, the use of a training sequence identifies those components of an original, analog waveform signal that must be removed or otherwise adjusted to correct for the effects of signal degradations. What remains is to actually remove or adjust such components by an appropriate, estimated amount. To do so, additional equalization or filtering is required.

A well-known technique for removing signal degradations, especially those caused by Inter-Symbol interference (ISI) due to channel insertion loss and reflections, or noise from crosstalk and other sources, is the non-linear, decision feedback equalizer (DFE). As is known in the art, a DFE uses proper values for so-called “tap coefficients” or feedback coefficients to effectively remove signal degradations. The values of the tap coefficients may be derived in part from a determined training sequence. The process of finding the proper values for “tap coefficients” in a DFE is referred to as an equalizer adaptation process. A DFE's tap coefficient values may be adapted to different values on different channels that cause different signal degradations. The behavior of a model DFE as set forth in the PCI Express or Peripheral Component Interconnect Express 3.0 specification (sometimes referred to as “PCIE 3.0” for short) is shown in FIG. 2. One form of a DFE that has been implemented in accordance with the model DFE shown in FIG. 2 to determine tap coefficients uses a so-called “exhaustive search” process. However, the inventor has found that this form of a DFE does not yield precise results and is computationally inefficient because it requires the analysis of many data points.

Accordingly, one objective of the present invention is to provide methods and systems that are more precise and computationally efficient in order to optimize the performance of DFEs used in a high-speed serial data link.

Another objective of the present invention is to provide methods and systems that optimize the performance of DFEs used in a high-speed serial data link by identifying optimal DFE tap values.

Yet another objective of the present invention is to provide methods and systems that optimize the performance of DFEs used in a high-speed serial data link by identifying optimal DFE tap values, where the optimized DFEs also comply with the behavior of a model DFE set forth in PCIE 3.0.

Other objectives and their related advantages provided by the present invention will be apparent from the text which follows along with the associated figures, read in conjunction with the appended claims.

SUMMARY

The present invention provides explicit DFE adaptation methods and related systems that find optimal DFE tap values for peak-to-peak based criteria.

Embodiments of the present invention may include methods and systems for optimizing the equalization of a high-speed data link comprising: a processor operable to execute stored instructions for, generating minimum and maximum voltage array input values derived from a sampled input signal; generating minimum and maximum voltage array output values representing a plurality of eye heights based on generated voltage array input values associated with a specified horizontal position of the unit interval and a feedback coefficient value: identifying a maximum voltage value representing a maximum eye height based on the generated voltage array output values; identifying one or more optimum feedback coefficient values based on the identified, maximum voltage value; and generating a non-linear, equalized output signal based on the one or more identified, optimum feedback coefficient values. The systems and methods may more particularly comprise or utilize a performance oscilloscope, such has a real-time oscilloscope or a sampling oscilloscope.

The data link being optimized may be a high speed serial data signal of at least 6 gigabits per second.

In additional embodiments of the invention, the methods and systems include the generation of a continuous, linearly equalized signal from the input signal prior to a DFE adaption process. In yet a further embodiment of the invention a performance oscilloscope for optimizing the equalization of a high-speed data link is provided that comprises a user interface for activating a DFE adaption process for a high speed serial data signal of at least 6 gigabits per second, and for indicating when a DFE adaption process, that identifies one or more optimum feedback coefficient values based on an identified, maximum voltage value associated with a specified horizontal position of the unit interval, is completed; and a display for displaying the user interface.

Embodiments of the invention may also include methods and systems system for optimizing the equalization of a high-speed data link including an input configured to receive an input waveform and a processor configured to perform decision feedback equalizer (DFE) adaptation on the input waveform for a multi-feedback coefficient DFE with n feedback coefficients, where n is greater than 1, The processor is operable to execute stored instructions, including measuring minimum and maximum voltage outputs associated with a specified horizontal position of the unit interval for n+1 bit patterns for the input waveform, formulating a linear equation using the measured minimum and maximum voltage outputs of the n+1 bit patterns, determining the optimal n feedback coefficients by solving the linear equation, and emulating the DFE to determine an equalized input waveform based on the determined optimal n feedback coefficients.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is representation of a high speed, serial data link system.

FIG. 2 is a representation of the behavior of a model, receiver based DFE in accordance with the PCIE 3.0 standard.

FIG. 3 depicts an eye height measurement specified by the PCIE 3.0 specification.

FIG. 4 depicts 2-bit sequence combinations in accordance with an embodiment of the present invention.

FIG. 5 depicts a measurement setup using an oscilloscope according to one embodiment of the present invention.

FIG. 6 depicts a functional block diagram of an oscilloscope according to one embodiment of the present invention.

FIG. 7 depicts a representation of minimum and maximum bit voltages as a function of a DFE feedback coefficient in accordance with an embodiment of the present invention.

FIG. 8 depicts a representation of eye height as a function of a DFE feedback coefficient in accordance with an embodiment of the present invention.

FIG. 9 depicts a flow diagram summarizing optimum DFE processes according to embodiments of the present invention

FIGS. 10A and 10B depict histograms both before and after DFE in accordance with an embodiment of the present invention.

FIG. 11 depicts a user interface that is a part of an oscilloscope for activating an optimum DFE process according an embodiment of the present invention.

FIG. 12 is a representation of the behavior of a model, receiver based DFE with multiple feedback coefficients.

FIG. 13 depicts another flow diagram summarizing optimum DFE processes according to embodiments of the present invention.

FIG. 14A depicts an eye diagram before a DFE process and FIG. 14B depicts an eye diagram after a DFE process.

DETAILED DESCRIPTION OF THE EMBODIMENTS

In accordance with the present invention, exemplary embodiments of DFEs are presented that utilize an explicit adaptation method, instead of an exhaustive search method, to identify optimal DFE tap or feedback coefficient values based on peak-to-peak based criteria. The explicit adaptation method provides a closed form solution (i.e., the optimal tap coefficient values are computed directly, not iteratively). Such DFEs comply with the behavior of a model DFE defined in the PCIE 3.0 specification.

Referring back to FIG. 2, the model, receiver-based DFE takes an analog output signal x_(k) from a continuous linear equalizer (CTLE), and outputs an equalized analog signal y_(k) and digital signal y*_(k), where: y _(k) =x _(k) −d ₁ sgn(y _(k-1))  (1) y* _(k) =sgn(y _(k)), |y* _(k)|=1  (2)

and where y_(k), may be modeled as a DFE, summed differential output voltage, y*_(k) as a decision function output voltage, x_(k) as a DFE differential input voltage, d₁ as a tap or feedback coefficient, and k as a sample index in UI. The PCIE 3.0 specification requires that a DFE adaption method identify an optimal value for the feedback coefficient d₁ so that the eye height V_(eye) and eye width of a signal, as displayed on an oscilloscope for example, are maximized. V_(eye) is measured from the maximum of bit 0's and minimum of bit 1's, evaluated at a specified horizontal position of the unit interval (UI), usually, around UI center, as illustrated in FIG. 3.

In accordance with embodiments of the present invention, DFE adaptation methods and systems are provided that identify optimal tap coefficient values that are associated with a maximized eye height measured at one specific horizontal position of the UI. In additional embodiments of the invention, once the optimal tap coefficient values are found that are associated with a maximized eye height, methods and systems may search the neighborhood around these tap coefficient values to identify optimal values that will maximize an eye area.

As is known in the art, the maximization of eye height uses a peak-to-peak based criteria. PCIE 3.0 specifies d₁ to be between in the range [−30 mV, 30 mV]. A known exhaustive search methodology mentioned briefly before analyzes data points throughout this range (referred to as the “allowed coefficient space”). To ensure a degree of accuracy the step size used to analyze all of the data points is small, which means that the exhaustive search method must check a lot of data points. This takes time. As a result, the exhaustive search method sacrifices computational speed for accuracy. For some applications, the sacrifice in computational speed is unacceptable. For example, the testing of some devices, etc., requires high speed computations and throughput, which translates into high speed, computationally efficient, DFE adaptation. For the reasons set forth above, the exhaustive search methodology was found to be inadequate by the present inventor.

Backtracking somewhat, as noted briefly before, a DFE uses a training sequence which may sometimes also be referred to as a “decision bit” sequence. Because of this a DFE may be characterized as using a “decision on previous bits” to determine a coefficient feedback value. As is known in the art, in accordance with PCIE 3.0 when optimal tap values are identified, the tap values remain unchanged unless the channel or transmitter is changed. Thus, in accordance with embodiments of the present invention once optimal tap values are identified they may be used to identify all of the components of an analog waveform input signal (or bits of its digitized version) that need to be adjusted to account for signal degradations. For example, in the model DFE shown in FIG. 2, the output y_(k) is determined by the input x_(k) and a decision on previous bit, denoted as y*_(k-1). U.S. Pat. No. 8,374,231 mentioned before sets forth methods and systems for identifying or determining a decision bit or training sequence, denoted y*_(k), for an input signal x_(k) using a performance oscilloscope, such as a sampling oscilloscope or real-time oscilloscope. For present purposes, it is assumed that a decision bit or training sequence y*_(k) has already been identified or determined. In accordance with an embodiment of the invention, the input signal can be associated with one of four 2-bit combinations, such as the combination denoted by points 6-9 in FIG. 4, according to the identified sequence. The combinations may then be used to generate and display a shifted eye diagram as shown, for example, in FIG. 4 on a performance oscilloscope.

Referring to FIG. 5 there is depicted a simplified illustration of a system 400, such as an oscilloscope, that includes elements for optimizing DFE in a high-speed serial data link according one embodiment of the present invention. It should be understood that all of the methods and processes discussed herein, and those set forth in U.S. Pat. No. 8,374,231 mentioned before may be implemented using an oscilloscope that includes substantially similar elements as the oscilloscope shown in FIG. 5. In embodiments of the invention the system 400 may comprise a performance oscilloscope, such as a real-time oscilloscope or sampling oscilloscope.

System 400 may, for example, take the form of one or more oscilloscopes designed and developed by Tektronix, Inc. The oscilloscope 400 may include multiple accessory interfaces 420 for connecting one or more accessory devices 440, such as differential probes designed and developed by Tektronix, Inc, to the oscilloscope 400. The accessory interfaces 420 may provide voltage power from the oscilloscope 400 to the accessory devices, which in the exemplary embodiment shown in FIG. 5 comprises a differential probe 440. The accessory interfaces may also provide bi-directional communications between the oscilloscope 400 and the probe 440. The probe 440 may be coupled via SMA coaxial cables 480 to a device under test (DUT) 460, such as a test fixture coupled to a SAS 6 G cable 500 acting as a channel in a high speed serial data link system.

The probe 440 may comprise a control box connected to the oscilloscope and a probe head coupled by a signal cable to the control box. The probe 440 may be coupled to the text fixture 460 using probing contacts.

The oscilloscope 400 may include a display device 520 which may include a graphical user interface for displaying signals processed by the oscilloscope 400 from DUT 460. Generally, oscilloscope 400 may include front panel controls 540, such as rotatable knobs, push buttons and the like for controlling the settings of the oscilloscope. Alternatively, the front panel controls may be graphically generated and presented as so called “soft keys” on the display section 520 so as to be accessed by a user of the oscilloscope 400.

It should be noted that although the channel 500 depicted in FIG. 5 is represented by cable 500, that the channel 500 may take the form of other mediums, such as copper wire, coaxial cable or may be simulated. Either way, the channel 500 includes impairments which degrade the signal being input into the DUT 460. These impairments must be removed and/or adjusted by the oscilloscope 400 in accordance with embodiments of the present invention. Alternatively, the DUT 460 may be removed and the oscilloscope 400 may be directly connected to the channel 500 (actual channel or simulated channel).

Viewed from another perspective, it should be understood that in one embodiment of the invention the oscilloscope 400 is an example of the reception means or receiver 16 depicted in FIG. 1. In an alternative embodiment, the oscilloscope 400 may also be placed before the channel 14 shown in FIG. 1 to, in effect, test the performance of the transmission means or transmitter 12, for example. In either case, the oscilloscope 400 may be connected to the channel 14 or transmission means 12 using an actual or simulated channel 500.

Referring now to FIG. 6, there is shown a representative block diagram of an oscilloscope according to one embodiment of the present invention. More particularly, shown is a block diagram of the oscilloscope 400 depicted in FIG. 5. The oscilloscope 400 may have separate signal channels 600 coupled to the accessory interfaces 420, two of which are represented in FIG. 6. Each signal channel 600 may have a separate acquisition means 620 that may include, for example, known electronic circuitry and/or devices for at least receiving an analog waveform input signal from the DUT 460 or channel 500 and converting the received signal into digitized samples. Each of the analog waveform input signals coupled to the signal channels 600 may also be coupled to trigger circuitry 640. The acquisition means 620 and the trigger circuitry 640 may be coupled to a programmable processing means 660 via a system bus 680. The system bus 680 may be further coupled to memory means 700 that may, for example, take the form of RAM, ROM and/or cache memory. RAM memory is operable to store volatile data, such as the digitized samples of the analog waveform input signal generated by the acquisition means 620. The system bus 680 may be further coupled to display circuitry 720 for controlling the display section 520 shown in FIG. 5, a mass storage unit or units 740, such as a hard disk drive, CD ROM drive, tape drive, floppy drive or the like that reads from and/or writes to appropriate mass storage media, and the front panel controls 540. It should be understood that any number of signal channels 600 may be included in the oscilloscope 400 with each channel having separate acquisitions means 620.

Executable instructions for implementing the equalization methods according to embodiments of the present invention and for otherwise controlling the oscilloscope 400 may be stored and accessed from memory means 700, more particularly, for example from ROM. Alternatively, the executable instructions may be stored and accessed from mass storage media of the mass storage unit 740 which in some embodiments may be included within memory means 700. The processing means 660 may be implemented as, for example, one or more programmable microprocessors, such as those designed and developed by Intel Corporation. The processing means 660 may also be implemented using multiple programmable controllers and/or one or more programmable digital signal processors. In yet another embodiment, when the processing means 660 is implemented using multiple controllers one may be used to control the acquisition and processing of the analog waveform input signal while the second may control the other operations of the oscilloscope 400. The oscilloscope 400 may be controlled using a Windows® Operating System, such a Windows® XP®, designed and developed by Microsoft, Corporation that is stored and accessed within one or more processors or controllers 660 and associated memory means 700.

The display circuitry 720 may include a display controller (not shown) for receiving instructions for controlling the display section 520 from processing means 660 and may receive data as well from a digital signal processor, for example, that is a part of processing means 660 for display by the display section 520. A bus controller (not shown) may also be included within the processing means 660 or included separately within the oscilloscope 400 for monitoring interfaces 420 and probes 440. The bus controller may also control communications between the probes 440 and the processing means 660 via communications bus 760. The bus 760 may comprise an I.sup.2C bus, IEEE 1494 bus, USB bus or the like, that provides bi-directional communications.

A power supply 780 may receive control signals from the processing means 660 for controlling the electrical power to the probes 440 via voltage lines 800 and the accessory interfaces 420.

Continuing, referring back to FIG. 4, as mentioned before embodiments of the present invention are directed at DFE adaptation methods and systems that maximize an eye height measured at one specific horizontal position of a UI, such as point 5 shown in FIG. 4. In particular, in one embodiment of the invention the processing means 660 may comprise a programmable digital signal processor (“processor” for short”), for example, which may be operable to access and execute instructions and associated data stored within memory means 700 in order to optimizing DFE in a high-speed serial data link by maximizing an eye height measured at one specific horizontal position of a UI. Such executable, stored instructions and data may be used by the processor 660 to select the specified horizontal position of the UI. In more detail, the processor 660 may be further operable to access executable instructions and data stored within memory means 700 in order to identify one or more voltages associated with the specified horizontal UI value and, thereafter, for generating the minimum and maximum voltage array input values (x values) from the identified voltages.

In accordance with an embodiment of the present invention, the eye height optimization problem can be derived as follows:

$\begin{matrix} \begin{matrix} {V_{eye} = {{\min\left( \left\{ y_{{bit}\; 1} \right\} \right)} - {\max\left( \left\{ y_{{bit}\; 0} \right\} \right)}}} \\ {= {\min\left( {\left\{ y_{{bit}\; 11} \right\},{\left\{ y_{{bit}\; 01} \right\} - {\max\left( {\left\{ y_{{bit}\; 10} \right\},\left\{ y_{{bit}\; 00} \right\}} \right)}}} \right.}} \\ {= {{\min\left( {{\min\left( \left\{ y_{{bit}\; 11} \right\} \right)},{\min\left( \left\{ y_{{bit}\; 01} \right\} \right)}} \right)} -}} \\ {\max\left( {{\max\left( \left\{ y_{{bit}\; 10} \right\} \right)},{\max\left( \left\{ y_{{bit}\; 00} \right\} \right)}} \right)} \end{matrix} & (3) \end{matrix}$ where {y_(bitxx)} denotes the voltage array measured at UI center for the last bits in a bit pattern specified by “xx”. For example y_(bit01) stands for the voltage measured for all of the 1 bits that have previous bits of 0. Note that from equation (1) and equation (2), the items in equation (3) can be written as:

$\begin{matrix} {\begin{matrix} {\mspace{79mu}{{\min\left( \left\{ y_{{bit}\; 11} \right\} \right)} = {{\min\left( {\left\{ x_{{bit}\; 11} \right\} - d_{1}} \right)} = {{\min\left( \left\{ x_{{bit}\; 11} \right\} \right)} - d_{1}}}}} \\ {= {x_{{bit}\; 11}^{\min} - d_{1}}} \end{matrix}{{\min\left( \left\{ y_{{bit}\; 01} \right\} \right)} = {{\min\left( {\left\{ x_{{bit}\; 01} \right\} + d_{1}} \right)} = {{{\min\left( \left\{ x_{{bit}\; 01} \right\} \right)} + d_{1}} = {x_{{bit}\; 01}^{\min} + d_{1}}}}}{{\max\left( \left\{ y_{{bit}\; 10} \right\} \right)} = {\max\left( {{\left\{ x_{{bit}\; 10} \right\} - d_{1}} = {{{\max\left( \left\{ x_{{bit}\; 10} \right\} \right)} - d_{1}} = {{x_{{bit}\; 10}^{\max} - {d_{1}{\max\left( \left\{ y_{{bit}\; 00} \right\} \right)}}} = {{\max\left( {\left\{ x_{{bit}\; 00} \right\} + d_{1}} \right)} = {{{\max\left( \left\{ x_{{bit}\; 00} \right\} \right)} + d_{1}} = {x_{{bit}\; 00}^{\max} + {d_{1}\mspace{79mu}{where}}}}}}}} \right.}}} & (4) \\ {\mspace{79mu}{x_{{bit}\; 11}^{\min} = {\min\left( \left\{ x_{{bit}\; 11} \right\} \right)}}} & (5) \end{matrix}$

It should be noted that the values x_(bit11) ^(min), x_(bit01) ^(min), x_(bit10) ^(max), x_(bit00) ^(max) in equation (5) are measured using the input signal x. Accordingly, in one embodiment of the present invention the processor 660 may be operable to access and execute stored instructions from the memory means 700 and data from memory means 700 to generate the minimum and maximum voltage array input values (x values above) derived from a sampled input signal (value x_(k) above).

It should be further noted that the sampled input signal may, or may not, have been output from a CTLE. If no CTLE is involved the input signal may be output from a channel or signal source.

The variables defined in equation (4) are linear functions of d₁. In accordance with embodiments of the present invention they may be represented as the thinner straight lines in FIG. 7.

The minimum of all 1 bits and the maximum of all 0 bits defined in equation (3) can be written as piece-wise linear function (the thicker lines in FIG. 7) as:

$\begin{matrix} {\begin{matrix} {{\min\left( \left\{ y_{{bit}\; 1} \right\} \right)} = {\min\left( {{\min\left( \left\{ y_{{bit}\; 11} \right\} \right)},{\min\left( \left\{ y_{{bit}\; 01} \right\} \right)}} \right)}} \\ {= \left\{ \begin{matrix} {{x_{{bit}\; 01}^{\min} + d_{1}},} & {d_{1} \leq {0.5\left( {x_{{bit}\; 11}^{\min} - x_{{bit}\; 01}^{\min}} \right)}} \\ {{x_{{bit}\; 11}^{\min} - d_{1}},} & {d_{1} > {0.5\left( {x_{{bit}\; 11}^{\min} - x_{{bit}\; 01}^{\min}} \right)}} \end{matrix} \right.} \end{matrix}\begin{matrix} {{\max\left( \left\{ y_{{bit}\; 0} \right\} \right)} = {\max\left( {{\max\left( \left\{ y_{{bit}\; 10} \right\} \right)},{\max\left( \left\{ y_{{bit}\; 00} \right\} \right)}} \right)}} \\ {= \left\{ \begin{matrix} {{x_{{bit}\; 10}^{\max} - d_{1}},} & {d_{1} \leq {0.5\left( {x_{{bit}\; 10}^{\max} - x_{{bit}\; 00}^{\max}} \right)}} \\ {{x_{{bit}\; 00}^{\max} + d_{1}},} & {d_{1} > {0.5\left( {x_{{bit}\; 10}^{\max} - x_{{bit}\; 00}^{\max}} \right)}} \end{matrix} \right.} \end{matrix}} & (6) \end{matrix}$

Accordingly, in an embodiment of the invention the processor 660 may be operable to access and execute stored instructions and data from the memory means 700 to generate minimum and maximum voltage array output values (y above) representing a plurality of eye heights based on the generated voltage array input values (x above) associated with a specified horizontal position of the unit interval and a feedback coefficient value (d₁).

As shown in FIG. 8, the eye height V_(eye), set forth in equation (3) can then be derived as function of d₁ based on FIG. 7 and equation (6). In an embodiment of the invention, this may be implemented using the processor 660 that is operable to access and execute stored instructions and data within memory means 700 for identifying a maximum voltage value representing a maximum eye height based on the generated voltage array output values.

In accordance with an embodiment of the present invention, the eye height function V_(eye) shown in FIG. 8 can be represented as an explicit piece-wise d function:

$\begin{matrix} {\begin{matrix} {V_{eye} = {{\min\left( \left\{ y_{{bit}\; 1} \right\} \right)} - {\max\left( \left\{ y_{{bit}\; 0} \right\} \right)}}} \\ {= \left\{ \begin{matrix} {{y^{\max} + {2\left( {d_{1} - d_{left}} \right)}},} & {d_{1} < d_{left}} \\ {y^{\max},} & {d_{left} \leq d_{1} \leq d_{right}} \\ {{y^{\max} + {2\left( {d_{right} - d_{1}} \right)}},} & {d_{1} > d_{right}} \end{matrix} \right.} \end{matrix}{where}{d_{left} = {\min\left( {{0.5\left( {x_{{bit}\; 11}^{\min} - x_{{bit}\; 01}^{\min}} \right)},{0.5\left( {x_{{bit}\; 10}^{\max} - x_{{bit}\; 00}^{\max}} \right)}} \right)}}{d_{right} = {\max\left( {{0.5\left( {{{x\_ bit11}\hat{}\min} - {{x\_ bit01}\hat{}\min}} \right)},{0.5\left( {{{x\_ bit10}\hat{}\max} - {{x\_ bit00}\hat{}\max}} \right)}} \right)}}} & (7) \\ {y^{\max} = {{0.5\left( {x_{{bit}\; 11}^{\min} + x_{{bit}\; 01}^{\min} - x_{{bit}\; 10}^{\max} - x_{{bit}\; 00}^{\max}} \right)} - \left( {d_{right} - d_{left}} \right)}} & (8) \end{matrix}$

An optimal value of d₁ with the consideration of the limits on d₁ may be explicitly computed based on the piece-wise linear equation (7). In accordance with an embodiment of the invention, this may be implemented using the processor 660 that is operable to access and execute instructions and data from memory means 700 in order to identify one or more optimum feedback coefficient values based on the identified, maximum voltage value.

Once the optimum feedback coefficient values are identified, in accordance with an embodiment of the invention the processor 660 may be further operable to access and execute stored instructions and data from memory means 700 in order to generate a non-linear, equalized output signal (y_(k)) based on the one or more identified, optimum feedback coefficient values. In sum, it can be said then that the processor 660, in effect, completes a DFE adaption process by identifying one or more optimum feedback coefficient values based on an identified, maximum voltage value, wherein the identified maximum voltage value is associated with a specified horizontal position of the UI by virtue of the inter-relationship of the UI and the steps involved in identifying the maximum voltage value.

FIG. 9 depicts a flow diagram summarizing the methods set forth above that optimize a DFE according to embodiments of the present invention. As shown in FIG. 9, an exemplary method may include one or more of the following:

identifying a bit sequence from digitized samples of the input signal if one has not already been identified, in steps 901, 902;

if CTLE is required, generating a continuous, linearly equalized signal from the input signal, in steps 903,904.

thereafter, generating minimum and maximum voltage array input values derived from a sampled input signal, in step 905;

generating minimum and maximum voltage array output values representing a plurality of eye heights based on generated voltage array input values associated with a specified horizontal position of the unit interval and a feedback coefficient value, in step 906;

identifying one or more voltages associated with the specified horizontal unit interval value; and generating the minimum and maximum voltage array input values from the identified voltages, in step 907;

identifying a maximum voltage value representing a maximum eye height based on the generated voltage array output values, in step 908;

identifying one or more optimum feedback coefficient values based on the identified, maximum voltage value, in step 909; and

generating a non-linear, equalized output signal based on the one or more identified, optimum feedback coefficient values, in step 910.

To further illustrate the concepts on the present invention, the following example is presented. A real-time oscilloscope, like system 400, acquires and converts an 8 Gb/s PCIE 3.0 analog input waveform signal x into digital signal samples using the acquisition means 620. The digitized signal samples may then be stored in memory means 700. Once digital signal samples are stored, the processor 660 may be operable to access executable instructions and the stored digitized samples in memory means 700 to complete post-processing on the samples. For example, the processor 660 may be operable to access the stored digitized signal samples of the input signal and executable instructions in memory means 700 in order to identify a training or bit sequence by, for example, using the process set forth in U.S. Pat. No. 8,374,231. Thereafter, post-processing may further include completing DFE adaption using a DFE adaptation module that is part of the processor 660.

It should be understood that instructions and data for implementing such a process may be stored in memory means 700. The processor 660 may, optionally, access executable instructions and data (e.g., digitized signal samples of waveform x) from memory means 700 to generate a continuous, linearly equalized signal using an optional CTLE module within the processor 660 prior to a DFE adaption process.

In accordance with an embodiment of the invention, the processor 660 may be operable to process the identified training or bit sequence to generate equalization taps that may be used to filter or equalize an analog input waveform signal from channel 500 in order to remove degradations caused by channel losses, reflections, cross talk, noise, and ISI.

In more detail, for example the processor 660 may be further operable to access executable instructions and data from memory means 700 in order to identify the minimum and maximum voltages for bits defined in equation (5), for example: x _(bit11) ^(min)=96.4 mV x _(bit01) ^(min)=34.3 mV x _(bit10) ^(max)=−41.5 mV x _(bit00) ^(max)=−99.7 mV

The processor 660 may then access executable instructions and data in memory means 700 for implementing the explicit piecewise linear function defined in equations (7) and (8) derived from these measurements. For example, if [−30 mV, 30 mV] is the limit of d₁, the processor 660 (e.g., DFE adaption module) may be operable to access executable instructions and data in memory means 700 in order to generate an optimal eye height voltage of V _(eye) =134.0 mV.

Based on this eye height voltage the processor 660 (again, for example, a DFE adaption module that is a part of processor 660) may, thereafter, be operable to access executable instructions and data in memory means 700 in order to compute d1, for example, as having a value in the range [29.1 mV, 30 mV].

It should be noted that, in comparison, if DFE equalization is not completed the eye height is V_(eye)=75.7 mV.

FIGS. 10A and B depict vertical histograms at UI center before and after DFE, respectively, as displayed on an oscilloscope such as system 400. As shown in these FIGUREs DFE adaption makes the eye open wider.

Turning to FIG. 11 there is depicted a user interface 900 that may be part of a performance oscilloscope for activating an optimum DFE process according an embodiment of the present invention. For example, the user interface 900 may be part of the display section 520. In accordance with an embodiment of the invention, an activation icon 920 may be displayed on the interface 900 which when clicked with a mouse, touched with a finger or otherwise activated begins the process of executing the DFE adaption process(es) for a high speed serial data signal of at least 6 gigabits per second described above. A save icon 1102 may be selected to (e.g., clicked, touched) to save the results of a DFE adaption process, for example in memory means 700. The user interface 900 may also include icon 1103 that may be displayed for indicating when the DFE adaption process described above (e.g., one that, in sum, identifies one or more optimum feedback coefficient values based on an identified, maximum voltage value associated with a specified horizontal position of the UI), is completed. It should be understood that the positioning of the icons 920, 1102 and 1103 is only for illustrative purposes and that their position may be altered without changing their function or the scope of the invention. Further, though shown as three separate icons one or more of the icons 920, 1102 and/or 1103 may be combined into as few as one icon (e.g., an icon that blinks at a certain rate depending on its function, or uses different colors depending on its function) or may be further separated into additional icons.

It should be noted that variations may be made to the details of the above-described embodiments of this invention without departing from the underlying principles thereof. For example, the DFE adaption process described above may be applied to both single and multi-tap DFEs. The methods and components described above for selecting the horizontal position of the unit interval and for generating a CTLE signal from the input signal apply equally to the below discussed methods using a multi-feedback coefficient DFE.

The behavior of a model DFE as set forth in the PCI Express or Peripheral Component Interconnect Express 4.0 specification (sometimes referred to as “PCIE 4.0” for short) increases the number of feedback coefficients or tap values from one to two. However, the below described embodiment may be used with any number of feedback coefficients. FIG. 12 depicts a DFE model with “n” number of feedback coefficients.

With an increase of using two feedback coefficient values, equations (1) and (2) above are rewritten as: y _(k) =x _(k) −d ₁ sgn(y _(k-1))−d ₂ sgn(y _(k-2))  (9) y* _(k) =sgn(y _(k)), |y* _(k)|=1  (10)

If more than two feedback coefficient values are utilized, such as n number of feedback coefficient values, then equation (9) above can be written as: y _(k) =x _(k) −d ₁ sgn(y _(k-1))−d ₂ sgn(y _(k-2)) . . . −d _(n) sgn(y _(k-n))  (11)

With a two-feedback coefficient DFE shown in equations (9) and (10), the DFE output y_(k) is determined based on the input signal x_(k) and the decision on the previous bits y*_(k-1) and y*_(k-2). As discussed above with respect to the one-feedback coefficient DFE, U.S. Pat. No. 8,374,231 teaches finding the decision bit sequence y*_(k) for input signal x_(k) on performance oscilloscopes, such as sampling oscilloscopes and performance real-time oscilloscopes. For present purposes, it is assumed that a decision bit or training sequence y*_(k) has already been identified or determined. The input signal waveform x_(k) can be labeled as one of eight 3-bit combinations according to a bit sequence of {x_(bit111)}, {x_(bit011)}, {x_(bit101)}, {x_(bit001)}, {x_(bit110)}, {x_(bit010)}, {x_(bit100)}, {x_(bit000)}.

The PCIE 4.0 specification requires that a DFE adaption method identify an optimal value for the feedback coefficients d₁ and d₂ so that the eye height V_(eye) multiplies with eye width, also called eye area, of a signal, as displayed on an oscilloscope for example, are maximized. V_(eye) is measured from the maximum of bit 0's and minimum of bit 1's, evaluated at a specified horizontal position of the unit interval (UI), usually, around the UI center, as discussed above and illustrated in FIG. 3.

In accordance with an embodiment of the present invention, the eye height optimization problem for a two-feedback coefficient DFE can be derived as follows:

                                          (12) $\begin{matrix} {V_{eye} = {{\min\left( \left\{ y_{{bit}\; 1} \right\} \right)} - {\max\left( \left\{ y_{{bit}\; 0} \right\} \right)}}} \\ {= {{{\min\left( {\left\{ y_{{bit}\; 111} \right\},\left\{ y_{{bit}\; 011} \right\},\left\{ y_{{bit}\; 101} \right\},\left\{ y_{{bit}\; 001} \right\}} \right)}\mspace{14mu}\ldots} -}} \\ {\max\left( {\left\{ y_{{bit}\; 110} \right\},\left\{ y_{{bit}\; 010} \right\},\left\{ y_{{bit}\; 100} \right\},\left\{ y_{{bit}\; 000} \right\}} \right)} \\ {= {{\min\left( {{\min\left( \left\{ y_{{bit}\; 111} \right\} \right)},{\min\left( \left\{ y_{{bit}\; 011} \right\} \right)},{\min\left( \left\{ y_{{bit}\; 101} \right\} \right)},{\min\left( \left\{ y_{{bit}\; 001} \right\} \right)}} \right)}\mspace{14mu}\ldots}} \\ {- {\max\left( {{\max\left( \left\{ y_{{bit}\; 110} \right\} \right)},{\max\left( \left\{ y_{{bit}\; 010} \right\} \right)},{\max\left( \left\{ y_{{bit}\; 100} \right\} \right)},{\max\left( \left\{ y_{{bit}\; 000} \right\} \right)}} \right.}} \end{matrix}$ where {y_(bitxxx)} denotes the voltage array measured at the UI center for the last bits in a bit pattern specified by “xxx”. For example y_(bit001) stands for the voltage measured for all of the 1 bits that have previous bits of 00. The following variables can be defined: x _(bit111) ^(min)=min({x _(bit111)})  (13) x _(bit011) ^(min)=min({x _(bit011)}) x _(bit101) ^(min)=min({x _(bit101)}) x _(bit001) ^(min)=min({x _(bit001)}) x _(bit110) ^(max)=max({x _(bit110)}) x _(bit010) ^(max)=max({x _(bit010)}) x _(bit100) ^(max)=max({x _(bit100)}) x _(bit000) ^(max)=max({x _(bit000)})

x_(bit111) ^(min), x_(bit011) ^(min), x_(bit101) ^(min), x_(bit001) ^(min), x_(bit110) ^(max), x_(bit010) ^(max), x_(bit100) ^(max), x_(bit000) ^(max) in equation (13) are measured based on the input signal x_(k). Equation (14) can be written using the terms of equation (12) with the variables defined in equation (13):

$\begin{matrix} {{{{\min\left( \left\{ y_{{bit}\; 111} \right\} \right)} = {x_{{bit}\; 111}^{\min} - d_{1} - d_{2}}}\min\left( \left\{ y_{{bit}\; 011} \right\} \right)} = {x_{{bit}\; 011}^{\min} - d_{1} + d_{2}}} & (14) \\ {\mspace{175mu}\vdots} & \; \\ {{\max\left( \left\{ y_{{bit}\; 000} \right\} \right)} = {x_{{bit}\; 000}^{\max} + d_{1} + d_{2}}} & \; \end{matrix}$

Equation (12) can be converted to a linear programming problem or equation to perform the optimization. Equation (15) shows a standard form of a linear program:

$\begin{matrix} {\min_{z}{f^{T}z\mspace{14mu}{such}\mspace{14mu}{that}\mspace{11mu}\left\{ \begin{matrix} {{A \cdot z} \leq b} \\ {{lb} \leq z \leq {ub}} \end{matrix}\; \right.}} & (15) \end{matrix}$ Where f, z, b, lb, and ub are vectors, and A is a matrix.

Equation (12) can then be converted into the linear format of (15) by following the below method. First, let z=[Veye d1 d2]^(T)  (16)

The four bit patterns bit111, bit011, bit101, bit001 all have the last bit being one, while the four bit patterns bit110, bit010, bit100, bit000 all have the last bit being zero. The maximum of every one of the four bit patterns bit110, bit010, bit100, bit000 will be subtracted from the minimum of every one of the four bit patterns bit111, bit011, bit101, bit001. These subtraction results should be greater or equal to the V_(eye) from equation (12). For example, consider bit111 and bit110, the inequality from equation (5) is Veye≦min({y _(bit111)})−max({y _(bit110)})  (17)

Using equations (14), (15), and (16) above, equation (17) can be rewritten as Veye−(x _(bit111) ^(min) −d ₁ −d ₂)+(x _(bit110) ^(max) −d ₁ −d ₂)≦0  (18)

Equation (18) can then be written as shown in equation (19): [1 0 0] z≦x _(bit111) ^(min) −x _(bit110) ^(max)  (19)

Similarly the bit111 and bit000 combination can be formulated as: [1 −2 −2] z≦x _(bit111) ^(min) −x _(bit000) ^(max)  (20)

This is done for all 4×4=16 combinations of minimum of bit 1 patterns and maximum of bit 0 patterns. Then A and b can be defined as:

$\begin{matrix} {{A = \begin{bmatrix} 1 & 0 & 0 \\ \vdots & \vdots & \vdots \\ 1 & {- 2} & {- 2} \end{bmatrix}}{b = \begin{bmatrix} {x_{{bit}\; 111}^{\min} - x_{{bit}\; 110}^{\max}} \\ \vdots \\ {x_{{bit}\; 111}^{\min} - x_{{bit}\; 000}^{\max}} \end{bmatrix}}} & (21) \end{matrix}$

The goal of maximizing vertical eye opening of Veye is equivalent to minimizing −Veye. So the objective vector f in (15) is f=[−1 0 0]^(T)  (22)

The limits on feedback coefficient values values can be put in Equation (15):

$\begin{matrix} {{{bl} = \begin{bmatrix} {- K} \\ d_{1}^{\min} \\ d_{2}^{\min} \end{bmatrix}}{{bu} = \begin{bmatrix} K \\ d_{1}^{\max} \\ d_{2}^{\max} \end{bmatrix}}} & (23) \end{matrix}$

Where K is a large number, for example, 1.0E6. For PCIE 4.0, the feedback coefficient limits are: [d ₁ ^(min) d ₁ ^(max)]=[−30 mV 30 mV] and [d ₂ ^(min) d ₂ ^(max)]=[−10 mV 10 mV]  (24)

Equations (21), (22), and (23) fully construct the DFE optimization problem into a standard linear programming problem format for equation (12). An efficient linear programming solver, such as Matlab®, is used to find the optimal DFE feedback coefficient values d₁ and d₂.

For the two-feedback coefficient DFE optimization problem above, the terms in equation (21) represents 16 inequalities. However, many of the 16 inequalities can be detected as trivial conditions so the actual computation complexity is lower.

As discussed above, equations (12), (21), (22), and (23) are derived based on a two-feedback coefficient DFE scenario. However, the same idea can be used for any multi-feedback coefficient DFE case, such as for a three-feedback coefficient case or a four-feedback coefficient case. For an n-feedback coefficient DFE, the combination of minimums of n+1 bit patterns ending with bit 1 and maximums of n+1-bit patterns ending with bit 0 will be total of 4^(n). As stated earlier, the actual computation complexity is lower since many of the inequalities can be detected as trivial conditions.

FIG. 13 depicts a flow diagram summarizing the methods set forth above that optimize a DFE according to embodiments of the present invention. Initially, a waveform is acquired in 1300 and then CTLE is performed in 1304, if required in 1302. In 1306, a clock recovery is performed and a bit sequence is identified from the digitized samples of the input signal.

The waveform is then resampled in 1308 according to the bit rate and the center of the UI is found in 1310. Real-time oscilloscopes usually sample waveforms at a fixed sample rate that is usually not synchronized to the signal bit rates. Then, in 1312, the minimums and maximums of n+1-bit patterns are measured, as set out in equation (13). The linear program problem is then constructed using equations (12), (21), (22), and (23) in 1314. The linear program problem is solved to find the optimal DFE feedback coefficients and the maximum vertical eye opening V_(eye) in 1316. After DFE adaptation is done in 1316, the DFE can be emulated with the fixed feedback coefficients to get the equalized waveform y_(k) in 1318.

Using a linear function, rather than exhaustive searching, as done in the prior art, allows for a much more efficient and accurate optimization of the DFE. A linear program solver can solve large scale linear program problems. Therefore, for DFEs with multi-feedback coefficient values, linear programming becomes even more useful as exhaustive searching grows exponentially with an increased number of feedback coefficients.

To further illustrate concepts of the present invention, the following example is presented. A real-time oscilloscope, like system 400, acquires and converts an 11.6 Gb/s high speed analog input waveform using the acquisition means 620. The digitized signal samples may then be stored in memory means 700. Once the digital signal samples are stored, the processor 660 may be operable to access executable instructions and the stored digitized samples in memory means 700 to complete post-processing on the samples. For example, the processor 660 may be operable to access the stored digitized samples of the input signal and executable instructions in memory means 700 to identify a training or bit sequence by, for example, using the process set forth in U.S. Pat. No. 8,374,231, discussed above. Thereafter, post-processing may further include completing DFE adaptation using a DFE adaptation module that is part of the processor 660.

As discussed above with respect to the one-feedback coefficient DFE adaptation, processor 660 may be operable to process the identified training or bit sequence to generate equalization feedback coefficients that may be used to filter or equalize an analog input waveform signal from channel 500 to remove degradations caused by channel losses, reflections, cross talk, and noise.

After CTLE emulation, the 11.6 Gb/s high speed waveform x is fed into a DFE adaptation module of processor 660. Without DFE, the eye height of the waveform is V_(eye)=7.8 mV.

The DFE adaptation module figures out the bit sequence after clock recovery. To begin with, a one-feedback coefficient DFE is considered. The measures of the minimum and maximum voltages on bits defined in equation (5) are: x _(bit11) ^(min)=39.0 mV x _(bit01) ^(min)=5.7 mV x _(bit10) ^(max)=−2.1 mV x _(bit00) ^(max)=−34.5 mV

Using the equations described above with respect to a one-feedback coefficient, the range of the one-feedback coefficient is: 1-tap DFE Range=[16.7 mV 16.2 mV] The optimal eye height with the one-feedback coefficient is V_(eye)=40.2 mV.

For a two-feedback coefficient, the measurements of the minimum and maximum voltages defined in (13) are: x _(bit111) ^(min)=43.2 mV x _(bit011) ^(min)=39.0 mV x _(bit101) ^(min)=9.5 mV x _(bit001) ^(min)=5.7 mV x _(bit110) ^(max)=−2.1 mV x _(bit010) ^(max)=−34.5 mV x _(bit000) ^(max)=−39.4 mV

Note that x_(bit11) ^(min)=min(x_(bit111) ^(min), x_(bit011) ^(min)).

Using the equations described above, after formulating the linear programming problem, and running Matlab®'s optimization tool function linprog to solve the linear program problem, the optimal two-feedback coefficient values are d₁=16.9 mV, d₂=2.5 mV.

The optimal eye height with the two-feedback coefficient DFE is V_(eye)=43.92 mV. This results in about a 10% increase in the vertical eye opening compared to a one-feedback coefficient DFE.

FIG. 14A depicts the eye opening before a DFE adaptation process, while FIG. 14B depicts the eye opening after the DFE adaptation process.

It should be noted that variations may be made to the details of the above-described embodiments of this invention without departing from the underlying principles thereof. The scope of the present invention should, therefore, be determined primarily by the following claims. 

I claim:
 1. A system for optimizing equalization of a high-speed data link comprising: an input configured to receive an input waveform; a processor configured to perform multi-feedback coefficient decision feedback equalization (DFE) adaptation on the input waveform using n feedback coefficients, where n is greater than 1, the processor configured to: measure minimum and maximum voltage outputs associated with a specified horizontal position of a unit interval for n+1 bit patterns for the input waveform; determine optimal n feedback coefficients using the measured minimum and maximum voltage outputs; and generate and output a non-linear, equalized signal by applying the determined optimal n feedback coefficients to the input waveform, wherein the optimal n feedback coefficients are determined by: $\min_{z}{f^{T}z\mspace{14mu}{such}\mspace{14mu}{that}\mspace{14mu}\left\{ \begin{matrix} {{A \cdot z} \leq b} \\ {{lb} \leq z \leq {ub}} \end{matrix} \right.}$ where f, z, b, lb, and ub are vectors and A is a matrix, wherein each of f, z, b, and A are determined using at least one of eye height, the n feedback coefficients, and the maximum and minimum voltage outputs for the n+1 bit patterns, and lb and ub are determined using maximum and minimum values of the n feedback coefficient values and a constant.
 2. The system as in claim 1, wherein the system further comprises a performance oscilloscope.
 3. The system as in claim 1, wherein the processor is further configured to select the specified horizontal position of the unit interval.
 4. The system as in claim 1, wherein the input waveform comprises a high speed serial data signal.
 5. The system as in claim 1, wherein the processor is further configured to identify one or more voltages associated with the specified horizontal unit interval value and to generate the minimum and maximum voltage outputs from the identified voltages.
 6. The system as in claim 1 further comprising: a user interface for activating the DFE, and for indicating when the DFE is completed; and a display for displaying the user interface.
 7. A method for optimizing equalization of a high-speed data link comprising: receiving an input waveform to be processed through a multi-feedback coefficient decision feedback equalizer (DFE) with n feedback coefficients, where n is greater than 1; measuring minimum and maximum voltage outputs associated with a specified horizontal position of a unit interval for n+1 bit patterns for the input waveform; determining optimal n feedback coefficients using the measured minimum and maximum voltage outputs; and generating and outputting a non-linear, equalized signal from the multi-feedback coefficient DFE by applying the determined optimal n feedback coefficients to the input waveform, wherein the optimal n feedback coefficients are determined by: $\min_{z}{f^{T}z\mspace{14mu}{such}\mspace{14mu}{that}\mspace{14mu}\left\{ \begin{matrix} {{A \cdot z} \leq b} \\ {{lb} \leq z \leq {ub}} \end{matrix} \right.}$ where f, z, b, lb, and ub are vectors and A is a matrix, wherein each of f, z, b, and A are determined using at least one of eye height, the n feedback coefficients, and the maximum and minimum voltage outputs for the n+1 bit patterns, and lb and ub are determined using maximum and minimum values of the n feedback coefficient values and a constant.
 8. The method as in claim 7, wherein the method is implemented using a performance oscilloscope and the DFE is part of the performance oscilloscope.
 9. The method as in claim 7, further comprising selecting the specified horizontal position of the unit interval.
 10. The method as in claim 7, wherein the input waveform comprises a high speed serial data signal.
 11. The method as in claim 7, further comprising: identifying one or more voltages associated with the specified horizontal unit interval value; and generating the minimum and maximum voltage outputs from the identified voltages. 